Solution
= Solution
The rational sequence $q_n=10^{-n}\lfloor10^n\sqrt2\rfloor$ satisfies $0\le\sqrt2-q_n<10^{-n}$. It is therefore a <Cauchy sequence> in the induced <metric> on $\mathbb Q$. Any limit in $\mathbb Q$ would also be its real limit, which is $\sqrt2\notin\mathbb Q$. Thus
$$
\boxed{\mathbb Q\text{ with the Euclidean metric is not complete}.}
$$
Completeness concerns limits lying in the space, so being a <dense subset> of $\mathbb R$ does not suffice.